Question

Let f(x) = 2x³ + 6x - 3 Use the limit definition of the derivative to calculate the derivative of f: ƒ'(x) = Use the same formula from above to calculate the derivative of this new function (i.e. the second derivative of f): f''(x) =____

Answer 1

The derivative of f(x) = 2x³ + 6x - 3 is f'(x) = 6x³ + 12x² + 6. The second derivative, f''(x), is 18x² + 24x.

Step-by-step explanation:

To calculate the derivative of the function f(x) = 2x³ + 6x - 3 using the limit definition of the derivative, we can follow these steps:

1. Start with the limit definition of the derivative: ƒ'(x) = lim[h→0] [(f(x + h) - f(x))/h]
2. Substitute the given function into the formula: ƒ'(x) = lim[h→0] [((2(x + h)³ + 6(x + h) - 3) - (2x³ + 6x - 3))/h]
3. Expand and simplify the expression within the limit: ƒ'(x) = lim[h→0] [((2x³ + 6x³ + 12x²h + 8xh² + 2h³ + 6x + 6h - 3) - (2x³ + 6x - 3))/h]
4. Combine like terms: ƒ'(x) = lim[h→0] [(6x³ + 12x²h + 8xh² + 2h³ + 6h)/h]
5. Cancel out the common factor of h: ƒ'(x) = lim[h→0] [6x³ + 12x² + 8xh + 2h² + 6]
6. Take the limit as h approaches 0: ƒ'(x) = 6x³ + 12x² + 6

Therefore, the derivative of f(x) = 2x³ + 6x - 3 is ƒ'(x) = 6x³ + 12x² + 6.

To find the second derivative (f''(x)), we can calculate the derivative of the first derivative using the same process. The result is:

f''(x) = 18x² + 24x